Section 5.1 Adding and Subtracting Polynomials
A polynomial is a type of algebraic expression with certain features we will describe below.
- A company’s sales, \(s\) (in millions of dollars), can be modeled by \(2.2t+5.8\text{,}\) where \(t\) stands for the number of years since \(2010\text{.}\)
- The height of an object from the ground, \(h\) (in feet), launched upward from the top of a building can be modeled by \(-16t^2+32t+300\text{,}\) where \(t\) represents the amount of time (in seconds) since the launch.
- The volume \(V\) (in cubic inches) of an open-top box with a square base can be calculated by \(30s^2-\frac{1}{2}s^3\text{,}\) where \(s\) stands for the length of the square base, and the box sides have to be cut from a certain square piece of metal.
All of the expressions above are polynomials. In this section, we will learn some basic vocabulary relating to polynomials and we’ll then learn how to add and subtract polynomials.
Subsection 5.1.1 Polynomial Vocabulary
We start this section with a flood of vocabulary terms associated with polynomials and examples of how to use them.
Definition 5.1.2.
A polynomial is an expression with one or more terms summed together. A term of a polynomial must either be a plain number or the product of a number and one or more variables raised to natural number powers. The expression \(0\) is also considered a polynomial, with zero terms.
Example 5.1.3.
- Here are three polynomials: \(x^2-5x+2\text{,}\) \(t^3-1\text{,}\) \(7y\text{.}\)
- The expression \(3x^4y^3+7xy^2-12xy\) is an example of a polynomial in more than one variable.
- The polynomial \(x^2-5x+3\) has three terms: \(x^2\text{,}\) \(-5x\text{,}\) and \(3\text{.}\)
- The polynomial \(3x^4+7xy^2-12xy\) also has three terms.
- The polynomial \(t^3-1\) has two terms.
A polynomial will never have a variable in the denominator of a fraction or under a radical.
Definition 5.1.4.
The coefficient (or numerical coefficient) of a term in a polynomial is the numerical factor in the term.
Example 5.1.5.
- The coefficient of the term \(\frac{4}{3}x^6\) is \(\frac{4}{3}\text{.}\)
- The coefficient of the second term of the polynomial \(x^2-5x+3\) is \(-5\text{.}\)
- The coefficient of the term \(\frac{y^7}{4}\) is \(\frac{1}{4}\text{.}\)
Checkpoint 5.1.6.
Identify which of the following are polynomials and which are not.
(a)
The expression \(-2x^9-\frac{7}{13}x^3-1\)
- is
- is not
a polynomial.
Explanation.
This is a polynomial.
(b)
The expression \(5x^{-2}-5x^2+3\)
- is
- is not
a polynomial.
Explanation.
This is not a polynomial because it has a negative exponent on a variable.
(c)
The expression \(\sqrt{2}x-\frac{3}{5}\)
- is
- is not
a polynomial.
Explanation.
This is a polynomial. Note that coefficients can have radicals even though variables cannot, and the square root here is only applied to the \(2\text{.}\)
(d)
The expression \(5x^3-5^{-5}x-x^4\)
- is
- is not
a polynomial.
Explanation.
This is a polynomial. Note that coefficients can have negative exponents even though variables cannot.
(e)
The expression \(\frac{25}{x^2}+23-x\)
- is
- is not
a polynomial.
Explanation.
This is not a polynomial because it has a variable in a denominator.
(f)
The expression \(37x^6-x+8^{\frac{4}{3}}\)
- is
- is not
a polynomial.
Explanation.
This is a polynomial. Note that coefficients can have fractional exponents even though variables cannot.
(g)
The expression \(\sqrt{7x}-4x^3\)
- is
- is not
a polynomial.
Explanation.
This is not a polynomial because it has a variable inside a radical.
(h)
The expression \(6x^{\frac{3}{2}}+1\)
- is
- is not
a polynomial.
Explanation.
This is not a polynomial because a variable has a fractional exponent.
(i)
The expression \(6^x-3x^6\)
- is
- is not
a polynomial.
Explanation.
This is not a polynomial because it has a variable in an exponent.
Definition 5.1.7.
A term in a polynomial with no variable factor is called a constant term.
Example 5.1.8.
The constant term of the polynomial \(x^2-5x+3\) is \(3\text{.}\)
Definition 5.1.9.
The degree of a term is one way to measure how “large” it is. When a term only has one variable, its degree is the exponent on that variable. When a term has more than one variable, its degree is the sum of the exponents on the variables. A constant term has degree \(0\text{.}\)
Example 5.1.10.
- The degree of \(5x^2\) is \(2\text{.}\)
- The degree of \(-\frac{4}{7}y^5\) is \(5\text{.}\)
- The degree of \(-4x^2y^3\) is \(5\text{.}\)
- The degree of \(17\) is \(0\text{.}\) (Constant terms have degree \(0\) .)
Definition 5.1.11.
The degree of a nonzero polynomial is the greatest degree that appears among its terms.
Definition 5.1.12.
The leading term of a polynomial is the term with the greatest degree (assuming there is no tie). The coefficient of a polynomial’s leading term is called the polynomial’s leading coefficient.
Example 5.1.13.
The degree of the polynomial \(4x^2-5x+3\) is \(2\) because the terms have degrees \(2\text{,}\) \(1\text{,}\) and \(0\text{,}\) respectively, and \(2\) is the largest. Its leading term is \(4x^2\text{,}\) and its leading coefficient is \(4\text{.}\)
To help us recognize a polynomial’s degree, the standard convention at this level is to write a polynomial’s terms in order from highest degree to lowest degree. When a polynomial is written in this order, it is written in standard form. For example, it is standard practice to write \(7-4x-x^2\) as \(-x^2-4x+7\) since \(-x^2\) is the leading term. By writing the polynomial in standard form, we can look at the first term to determine both the polynomial’s degree and leading term.
There are special names for polynomials with a small number of terms, and for polynomials with certain degrees.
- monomial
- A polynomial with one term, such as \(3x^5\text{,}\) is called a monomial.
- binomial
- A polynomial with two terms, such as \(3x^5+2x\text{,}\) is called a binomial.
- trinomial
- A polynomial with three terms, such as \(x^2-5x+3\text{,}\) is called a trinomial.
- constant polynomial
- A zero-degree polynomial is called a constant polynomial. An example is the polynomial \(7\text{,}\) which has degree zero.
- linear polynomial
- A first-degree polynomial is called a linear polynomial. An example is \(-2x+7\text{.}\)
- quadratic polynomial
- A second-degree polynomial is called a quadratic polynomial. An example is \(4x^2-2x+7\text{.}\)
- cubic polynomial
- A third-degree polynomial is called a cubic polynomial. An example is \(x^3+4x^2-2x+7\text{.}\)
Fourth-degree and fifth-degree polynomials are called quartic and quintic polynomials, respectively. If the degree of the polynomial, \(n\text{,}\) is greater than five, we’ll simply call it an \(n\)th-degree polynomial. For example, the polynomial \(5x^8-4x^5+1\) is an \(8\)th-degree polynomial.
Subsection 5.1.2 Adding and Subtracting Polynomials
Example 5.1.14. Production Costs.
Bayani started a company that makes one product: one-gallon ketchup jugs for industrial kitchens. The company’s production expenses only come from two things: supplies and labor. The cost of supplies, \(S\) (in thousands of dollars), can be modeled by \(S=0.05x^2+2x+30\text{,}\) where \(x\) is number of thousands of jugs of ketchup produced. The labor cost for his employees, \(L\) (in thousands of dollars), can be modeled by \(0.1x^2+4x\text{,}\) where \(x\) again represents the number of jugs they produce (in thousands of jugs). Find a model for the company’s total production costs.
Since Bayani’s company only has these two costs, we can find a model for the total production costs, \(C\) (in thousands of dollars), by adding the supply costs and the labor costs:
\begin{equation*}
C=\left(0.05x^2+2x+30\right)+\left(0.1x^2+4x\right)
\end{equation*}
To finish simplifying our total production cost model, we’ll combine the like terms:
\begin{align*}
C \amp= 0.05x^2+0.1x^2+2x+4x+30\\
\amp= 0.15x^2+6x+30
\end{align*}
This simplified model can now calculate Bayani’s total production costs \(C\) (in thousands of dollars) when the company produces \(x\) thousand jugs of ketchup.
In short, the process of adding two or more polynomials involves recognizing and then combining the like terms.
Checkpoint 5.1.15.
Add the polynomials.
\(\displaystyle{\left({6x^{2}+4x}\right)+\left({4x^{2}-5x}\right)}\)
Explanation.
We combine like terms as follows
\begin{equation*}
\begin{aligned}
\left({6x^{2}+4x}\right)+\left({4x^{2}-5x}\right)\amp = \left({6x^{2}+4x^{2}} \right)+\left({4x-5x} \right)\\
\amp = {10x^{2}-x}
\end{aligned}
\end{equation*}
Example 5.1.16.
Simplify the expression \(\left(\frac{1}{2}x^2-\frac{2}{3}x-\frac{3}{2}\right)+\left(\frac{3}{2}x^2+\frac{7}{2}x-\frac{1}{4}\right)\text{.}\)
Explanation.
\begin{align*}
\amp\left(\frac{1}{2}x^2\firsthighlight{{}-\frac{2}{3}x}\secondhighlight{{}-\frac{3}{2}}\right)+\left(\frac{3}{2}x^2\firsthighlight{{}+\frac{7}{2}x}\secondhighlight{{}-\frac{1}{4}}\right)\\
\amp=\left(\frac{1}{2}x^2+\frac{3}{2}x^2\right)+\left(\firsthighlight{-\frac{2}{3}x+\frac{7}{2}x}\right)+\left(\secondhighlight{-\frac{3}{2}+\left(-\frac{1}{4}\right)}\right)\\
\amp= \left(\frac{4}{2}x^2\right)+\left(\firsthighlight{-\frac{4}{6}x+\frac{21}{6}x}\right)+\left(\secondhighlight{-\frac{6}{4}+\left(-\frac{1}{4}\right)}\right)\\
\amp= \left(2x^2\right)+\firsthighlight{\frac{17}{6}x}+\left(\secondhighlight{-\frac{7}{4}}\right)\\
\amp=2x^2\firsthighlight{{}+\frac{17}{6}x}\secondhighlight{{}-\frac{7}{4}}
\end{align*}
Example 5.1.17. Profit, Revenue, and Costs.
From Example 14, we know Bayani’s ketchup company’s production costs, \(C\) (in thousands of dollars), for producing \(x\) thousand jugs of ketchup is modeled by \(C=0.15x^2+6x+30\text{.}\) The revenue, \(R\) (in thousands of dollars), from selling the ketchup can be modeled by \(R=13x\text{,}\) where \(x\) stands for the number of thousands of jugs of ketchup sold. The company’s net profit can be calculated using the concept:
\begin{equation*}
\text{net profit} = \text{revenue} - \text{costs}
\end{equation*}
Assuming all products produced will be sold, a polynomial to model the company’s net profit, \(P\) (in thousands of dollars) is:
\begin{align*}
P \amp= R-C\\
\amp= \left(13x\right)-\left(0.15x^2+6x+30\right)\\
\amp= 13x-0.15x^2-6x-30\\
\amp= -0.15x^2+\left(13x+(-6x)\right)-30\\
\amp=-0.15x^2+7x-30
\end{align*}
The key distinction between the addition and subtraction of polynomials is that when we subtract a polynomial, we must subtract each term in that polynomial.
Notice that our first step in simplifying the expression in Example 17 was to subtract every term in the second expression. We can also think of this as distributing a factor of \(-1\) across the second polynomial, \(0.15x^2+6x+30\text{,}\) and then adding these terms as follows:
\begin{align*}
P \amp= R-C\\
\amp= \left(13x\right)-\left(0.15x^2+6x+30\right)\\
\amp= 13x+(-1)(0.15x^2)+(-1)(6x)+(-1)(30)\\
\amp= 13x-0.15x^2-6x-30\\
\amp= -0.15x^2+\left(13x+(-6x)\right)-30\\
\amp=-0.15x^2+7x-30
\end{align*}
Example 5.1.18.
Subtract \(\left(5x^3+4x^2-6x\right)-\left(-3x^2+9x-2\right)\text{.}\)
Explanation.
We must first subtract every term in \(\left(-3x^2+9x-2\right)\) from \(\left(5x^3+4x^2-6x\right)\text{.}\) Then we can combine like terms.
\begin{align*}
\amp\left(5x^3+4x^2-6x\right)-\left(-3x^2+9x-2\right)\\
\amp= 5x^3+4x^2-6x \highlight{{}+{}} 3x^2 \highlight{{}-{}} 9x \highlight{{}+{}} 2\\
\amp= 5x^3+\left(4x^2+3x^2\right)+\left(-6x+(-9x)\right)+2\\
\amp= 5x^3+7x^2-15x+2
\end{align*}
Checkpoint 5.1.19.
Subtract the polynomials.
\(\displaystyle{\left({3x-10}\right)-\left({-5x+7}\right)}\)
Explanation.
We combine like terms as follows
\(\begin{aligned}
\left({3x-10}\right)-\left({-5x+7}\right)\amp = \left(3 x-(-5 x)\right)+\left(-10-7\right)\\
\amp = {8x-17}
\end{aligned}\)
Let’s look at one last example where the polynomial has multiple variables. Remember that like terms must have the same variable(s) with the same exponent.
Example 5.1.20.
Subtract \(\left( 3x^2y+8xy^2-17y^3 \right)-\left(2x^2y+11xy^2+4y^2 \right)\text{.}\)
Explanation.
Again, we’ll begin by subtracting each term in \(\left(2x^2y+11xy^2+4y^2\right)\text{.}\) Once we’ve done this, we’ll need to identify and combine like terms.
\begin{align*}
\amp\left( 3x^2y+8xy^2-17y^3 \right)-\left(2x^2y+11xy^2+4y^2\right)\\
\amp=
3x^2y+8xy^2-17y^3 \highlight{{}-{}} 2x^2y \highlight{{}-{}} 11xy^2 \highlight{{}-{}} 4y^2\\
\amp=\left(3x^2y-2x^2y\right)+\left(8xy^2-11xy^2\right)+\left(-17y^3-4y^2\right)\\
\amp= x^2y-3xy^2-17y^3-4y^2
\end{align*}
Subsection 5.1.3 Evaluating Polynomial Expressions
Evaluating expressions was introduced in Section 1.1, and involves replacing the variable(s) in an expression with specific numbers and calculating the result.
Example 5.1.21.
Evaluate the expression \(C=0.15x^2+6x+30\) from Example 14 for \(x=10\) and explain what this means in context.
Explanation.
We will replace \(x\) with \(10\text{:}\)
\begin{align*}
C\amp=0.15x^2+6x+30\\
\amp=0.15(\substitute{10})^2+6(\substitute{10})+30\\
\amp=0.15(100)+60+30\\
\amp=15+90\\
\amp=105
\end{align*}
The context was that \(x\) represents so many thousands of jugs of ketchup, and \(C\) represents the total cost, in thousands of dollars, to produce that many jugs. So in context, we can interpret this as it costing \(\$105{,}000\) to produce \(10{,}000\) jugs of ketchup.
Recall that in Subsection 1.1.4 and Example 1.1.13 we discussed how \((-2)^2\) and \(-2^2\) are not the same expressions. The first expression, \((-2)^2\text{,}\) represents the number \(-2\) squared, and is \((-2)(-2)=4\text{.}\) The second expression, \(-2^2\text{,}\) is the opposite of the number that you get after you square \(2\text{,}\) and is \(-2^2=-(2\cdot 2) = -4\text{.}\)
Example 5.1.22.
Evaluate the expression \(-12y^3+4y^2-9y+2\) for \(y=-5\text{.}\)
Explanation.
We will replace \(y\) with \(-5\) and simplify the result:
\begin{align*}
-12y^3+4y^2-9y+2 \amp= -12(\substitute{-5})^3+4(\substitute{-5})^2-9(\substitute{-5})+2\\
\amp= -12(-125)+4(25)+45+2\\
\amp= 1647
\end{align*}
Reading Questions 5.1.4 Reading Questions
1.
What are the names for a polynomial with one term? With two terms? With three terms?
2.
Adding and subtracting polynomials is mostly about combining terms.
3.
What should you be careful with when evaluating a polynomial for a negative number?
Exercises 5.1.5 Exercises
Review and Warmup
Identifying Terms.
List the terms in each expression.
1.
\({13V-85d-7.4v}\)
2.
\({{\frac{3}{10}}D-4z+8.9D}\)
3.
\({-83x^{9}+{\frac{1}{6}}x^{7}-x^{2}}\)
4.
\({{\frac{2}{9}}t^{3}-1.5t^{6}+0.5t^{3}}\)
Combining Like Terms.
Simplify the expression by combining like terms if possible.
5.
\({2.3C+38C}\)
6.
\({{\frac{10}{3}}I - {\frac{1}{5}}I}\)
7.
\({N^{6}+63N^{4}+5.6N^{6}+1.3N^{4}}\)
8.
\({{\frac{3}{4}}U^{5} - {\frac{1}{5}}U^{5} - {\frac{4}{9}}U^{4}+U^{4}}\)
Vocabulary
Exercise Group.
What specific type of polynomial is the given polynomial? What is its degree? What is its leading term? What is its leading coefficient?
9.
\({9t^{4}-7t}\)
10.
\({-8t^{8}+6t^{9}}\)
11.
\({6y^{3}}\)
12.
\({{\frac{1}{2}}x^{7}}\)
13.
\({-12.8+49.5z^{2}-83.2z^{5}}\)
14.
\({9.4z^{8}-20.1z+0.2z^{9}}\)
15.
\({3z^{9}-9z^{4}t^{4}}\)
16.
\({5t^{3}+3t^{2}y^{9}}\)
17.
\({{\frac{7}{9}}yt^{4}+{\frac{3}{8}}y^{8}t^{5} - {\frac{5}{2}}y^{4}t^{8}}\)
18.
\({{\frac{9}{4}}yt^{8}+{\frac{9}{8}}y^{3}t - {\frac{3}{4}}y^{8}t^{7}}\)
Simplifying Polynomials
Exercise Group.
Add the polynomials.
19.
\(\left({3x+1}\right)+\left({-3x-9}\right)\)
20.
\(\left({-4x-9}\right)+\left({-6x-8}\right)\)
21.
\(\left({7y^{9}-9y^{4}}\right)+\left({-2y^{9}-y^{4}}\right)\)
22.
\(\left({2y^{6}-9y^{8}}\right)+\left({8y^{6}+y^{8}}\right)\)
23.
\(\left({-4r^{4}+9r^{3}-2}\right)+\left({3r^{4}-6r^{3}-5}\right)\)
24.
\(\left({-9r^{9}+9r^{6}+7}\right)+\left({3r^{9}+2r^{6}-1}\right)\)
25.
\(\left({-{\frac{1}{8}}t+{\frac{5}{9}}}\right)+\left({-{\frac{1}{3}}t - {\frac{8}{7}}}\right)\)
26.
\(\left({-{\frac{5}{9}}t+1}\right)+\left({t+{\frac{2}{5}}}\right)\)
27.
\(\left({{\frac{7}{8}}t^{2} - {\frac{3}{4}}t - {\frac{2}{9}}}\right)+\left({{\frac{7}{9}}t^{2}+3t+1}\right)\)
28.
\(\left({-{\frac{1}{3}}x^{3}+x-1}\right)+\left({-{\frac{2}{3}}x^{3}+2x+{\frac{1}{3}}}\right)\)
29.
\(\left({-{\frac{6}{5}}x^{5}+{\frac{3}{8}}x^{7}+{\frac{1}{3}}x^{9}}\right)+\left({-{\frac{4}{7}}x^{5} - {\frac{1}{2}}x^{9}}\right)\)
30.
\(\left({-y^{2} - {\frac{1}{4}}y^{7} - {\frac{8}{5}}y^{9}}\right)+\left({4y^{2} - {\frac{1}{4}}y^{9}}\right)\)
31.
\(\left({8y^{5}-3y^{6}}\right)+\left({-4y^{5}+y}\right)\)
32.
\(\left({7r^{2}+8r}\right)+\left({-2r^{2}+6r^{8}}\right)\)
33.
\(\left({-2.2r^{8}+0.7r^{4}-3.7r^{5}}\right)+\left({-3.6r^{5}+3.6r^{8}+1.7r^{9}}\right)\)
34.
\(\left({7.7t^{5}+2.3t^{8}+4.5t^{2}}\right)+\left({2.4t^{2}-7.5t^{5}-5.8t^{9}}\right)\)
Exercise Group.
Subtract the polynomials.
35.
\(\left({-8t-4}\right)-\left({9t+6}\right)\)
36.
\(\left({5t+3}\right)-\left({4t+7}\right)\)
37.
\(\left({-2x^{4}+6x}\right)-\left({-3x^{4}+8x}\right)\)
38.
\(\left({-7x+6x^{6}}\right)-\left({8x+9x^{6}}\right)\)
39.
\(\left({7y^{9}+5y^{7}-4}\right)-\left({-9y^{9}+4y^{7}-7}\right)\)
40.
\(\left({2y^{4}+6y^{3}+7}\right)-\left({-7y^{4}-6y^{3}-2}\right)\)
41.
\(\left({-{\frac{9}{5}}r - {\frac{3}{5}}}\right)-\left({{\frac{3}{5}}r+{\frac{1}{5}}}\right)\)
42.
\(\left({-{\frac{1}{2}}r - {\frac{9}{5}}}\right)-\left({-{\frac{7}{3}}r - {\frac{9}{8}}}\right)\)
43.
\(\left({-{\frac{3}{2}}t^{3}+{\frac{5}{4}}t^{2}+{\frac{3}{2}}}\right)-\left({{\frac{2}{3}}t^{3} - {\frac{6}{7}}t^{2} - {\frac{5}{7}}}\right)\)
44.
\(\left({-t^{2}+{\frac{4}{7}}t+{\frac{5}{3}}}\right)-\left({-{\frac{3}{5}}t^{2}-5t+{\frac{1}{4}}}\right)\)
45.
\(\left({-2t^{2} - {\frac{5}{4}}t^{5} - {\frac{3}{2}}t^{9}}\right)-\left({-{\frac{2}{3}}t^{2}-2t^{9}}\right)\)
46.
\(\left({-{\frac{2}{3}}x^{5}+2x^{7}+{\frac{7}{4}}x^{8}}\right)-\left({-{\frac{1}{2}}x^{5}-6x^{8}}\right)\)
47.
\(\left({4x^{9}-4x^{3}}\right)-\left({6x^{9}-9x^{6}}\right)\)
48.
\(\left({4y^{6}+7y^{7}}\right)-\left({6y^{6}-3y^{3}}\right)\)
49.
\(\left({-3.1y^{3}+9y+6.3y^{2}}\right)-\left({-6.5y^{2}+8.3y^{3}+3.2y^{7}}\right)\)
50.
\(\left({6.8r^{8}-9.1r^{5}-5.4r^{7}}\right)-\left({-0.4r^{7}-2.5r^{8}-4.1r^{4}}\right)\)
Exercise Group.
Add or subtract the multivariable polynomials as indicated.
51.
\(\left({8r+z}\right)+\left({3r-4z}\right)\)
52.
\(\left({-3r-5x}\right)+\left({3r+7x}\right)\)
53.
\(\left({-4t^{7}r^{6}-tr^{3}}\right)-\left({-2t^{7}r^{6}+5tr^{3}}\right)\)
54.
\(\left({-2t^{2}y^{6}+7t^{7}y^{8}}\right)-\left({4t^{2}y^{6}-7t^{7}y^{8}}\right)\)
55.
\(\left({-x^{6}-5x^{4}y^{6}+9y^{3}}\right)+\left({3x^{6}-2x^{4}y^{6}+8y^{3}}\right)\)
56.
\(\left({2x^{3}+4xt^{6}-5t^{8}}\right)+\left({-8x^{3}-7xt^{6}+9t^{8}}\right)\)
57.
\(\left({-{\frac{1}{7}}y - {\frac{1}{3}}z}\right)-\left({-{\frac{3}{8}}z+2y}\right)\)
58.
\(\left({{\frac{7}{2}}z+{\frac{1}{6}}x}\right)-\left({{\frac{5}{8}}x-3z}\right)\)
59.
\(\left({{\frac{5}{3}}z^{3}t^{5} - {\frac{1}{2}}zt^{8} - {\frac{7}{5}}t^{8}}\right)+\left({-{\frac{1}{9}}z^{3}t^{5} - {\frac{7}{8}}zt^{8} - {\frac{7}{2}}t^{7}}\right)\)
60.
\(\left({-{\frac{3}{8}}r^{3}y^{5}+{\frac{3}{4}}r^{2}y^{3} - {\frac{1}{2}}y^{3}}\right)+\left({{\frac{2}{7}}r^{3}y^{5} - {\frac{3}{2}}r^{2}y^{3} - {\frac{5}{7}}y^{2}}\right)\)
61.
\(\left({-{\frac{8}{5}}r^{2}x^{3}+{\frac{4}{9}}r^{3}x^{8}+{\frac{2}{3}}r^{8}x^{7}}\right)-\left({{\frac{3}{5}}r^{2}x^{3} - {\frac{5}{6}}r^{8}x^{7}}\right)\)
62.
\(\left({-{\frac{6}{7}}tz^{3}+{\frac{8}{7}}t^{4}z^{9}+{\frac{6}{7}}t^{5}z}\right)-\left({{\frac{6}{5}}tz^{3}+{\frac{4}{5}}t^{5}z}\right)\)
63.
\(\left({-2t^{9}y^{3}-5t^{2}y^{2}}\right)+\left({t^{9}y^{3}+6t^{5}y^{6}}\right)\)
64.
\(\left({5x^{4}y^{8}+4x^{9}y^{3}}\right)+\left({-8x^{4}y^{8}+6x^{5}y^{9}}\right)\)
65.
\(\left({5x^{7}r^{4}+4.7x^{6}r^{3}-1.4x^{4}}\right)-\left({-1.9x^{4}-5.5x^{7}r^{4}+0.3x^{2}r^{5}}\right)\)
66.
\(\left({-2.4y^{2}z^{5}+4.3y^{5}z^{7}-3.6y^{6}}\right)-\left({-0.8y^{6}+6.6y^{2}z^{5}-1.2y^{8}z^{6}}\right)\)
Evaluating Polynomials
Exercise Group.
Evaluate the given polynomial at the given numbers.
67.
\({-{\frac{1}{4}}y+5}\)
for \(y={-{\frac{5}{3}}}\text{,}\) \({{\frac{2}{5}}}\text{,}\) and \({1}\)
68.
\({5y+1}\)
for \(y={{\frac{4}{5}}}\text{,}\) \({1}\text{,}\) and \({3}\)
69.
\({4r^{2}+3r+2}\)
for \(r=2\text{,}\) \(3\text{,}\) and \(5\)
70.
\({r^{2}+4r+5}\)
for \(r=2\text{,}\) \(5\text{,}\) and \(4\)
71.
\({-5z^{2}-3z+1}\)
for \(z=-7\text{,}\) \(-3\text{,}\) and \(3\)
72.
\({2z^{2}+3z-3}\)
for \(z=-7\text{,}\) \(-3\text{,}\) and \(3\)
73.
\({-x^{2}-5x+4}\)
for \(x=-1.6\text{,}\) \(-2.3\text{,}\) and \(-4.2\)
74.
\({-4x^{2}-x+1}\)
for \(x=-1.7\text{,}\) \(2.8\text{,}\) and \(-3.3\)
75.
\({4t^{2}+5t-4}\)
for \(t={{\frac{1}{2}}}\text{,}\) \({-{\frac{3}{2}}}\text{,}\) and \({1}\)
76.
\({2y^{2}-2y+4}\)
for \(y={-{\frac{3}{2}}}\text{,}\) \({-3}\text{,}\) and \({{\frac{4}{3}}}\)
77.
\({-y^{6}-3y^{3}}\)
for \(y=-2\text{,}\) \(2\text{,}\) and \(1\)
78.
\({-r^{2}+3r}\)
for \(r=1\text{,}\) \(2\text{,}\) and \(-2\)
Applications
Falling Object.
The polynomial \(\frac12gt^2+v_0t+h_0\) gives the height of an object at time \(t\) (measured in seconds) when it is thrown straight up with an initial velocity \(v_0\) (measured in feet per second) from an initial height \(h_0\) (measured in feet) in a place where the acceleration from gravity is \(g\) (measured in feet per second per second). The acceleration of gravity on Earth is \(-32\,\frac{\mathrm{ft}}{\mathrm{s}^2}\text{.}\)
79.
What is the height of an object on Earth \(2.5\) seconds after it is thrown straight up with an initial speed of \(70\) feet per second from an initial height of \(6\) feet?
80.
What is the height of an object on Earth \(2.7\) seconds after it is thrown straight up with an initial speed of \(60\) feet per second from an initial height of \(4\) feet?
Sum of Consecutive Integers.
It turns out that if you add up all the whole numbers \(1+2+3+\cdots+n\text{,}\) the result is the polynomial \(\frac12n^2+\frac12n\text{.}\)
81.
What is the result from adding \(1+2+3+\cdots+983\text{?}\)
82.
What is the result from adding \(1+2+3+\cdots+538\text{?}\)
Sum of Consecutive Squares.
It turns out that if you add up all the square numbers \(1^2+2^2+3^2+\cdots+n^2\text{,}\) the result is the polynomial \(\frac13n^3 + \frac12n^2 + \frac16n\text{.}\)
83.
What is the result from adding \(1^2+2^2+3^2+\cdots+118^2\text{?}\)
84.
What is the result from adding \(1^2+2^2+3^2+\cdots+130^2\text{?}\)
Sales and Costs.
A company that sells a certain product can use polynomials to model its sales revenue (in dollars) and its costs (also in dollars). The variable \(x\) represents the number of units produced. Profit is revenue minus costs.
85.
If revenue is modeled by \({4.6x^{2}+1.68x+2218}\) and costs are modeled by \({3x^{2}+1.5x+1936}\text{,}\) what is the model for profit?
86.
If revenue is modeled by \({3.6x^{2}+5.3x+4704}\) and costs are modeled by \({2.8x^{2}+3.59x+1871}\text{,}\) what is the model for profit?
Algae.
There are two types of algae (blue and green) growing in a pond, and each type’s total biomass (measured in kilograms) can be modeled using a polynomial where the variable \(t\) represents time.
87.
If the blue algae biomass is modeled by \({0.001x^{2}+0.14x+124}\) and the green algae biomass is modeled by \({0.006x^{2}+0.79x+116}\text{,}\) what is a model for the total algae biomass?
88.
If the blue algae biomass is modeled by \({0.006x^{2}-0.36x+174}\) and the green algae biomass is modeled by \({0.007x^{2}-0.27x+114}\text{,}\) what is a model for the total algae biomass?
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